On the generalized Lie structure of associative algebras |
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Authors: | Y. Bahturin D. Fischman S. Montgomery |
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Affiliation: | (1) Faculty of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russian Federation;(2) Department of Mathematics, California State University, 92407 San Bernardino, CA, USA;(3) Department of Mathematics, University of Southern California, 90089-1113 Los Angeles, CA, USA |
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Abstract: | We study the structure of Lie algebras in the category H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain. Dedicated to the memory of S. A. Amitsur Supported by a Fulbright grant. Supported by NSF grant DMS-9203375. |
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